Saturday, December 13, 2014

Changing the Culture:

I stumbled upon this article recently and it made me think of how math is thought about within the family unit.
How do we did get a classroom, school, community, entire society to all share the same cultural values in regards to the importance of math? 

The importance to literacy seems to be there.  Parent/teacher conferences will sometimes often have a similar theme if the parent acknowledges during their time in school they were not a strong student.  The parent will realize the importance of their child reading fluently and having comprehension of the text.  No matter what challenges the parent has in providing support to their child they are on board with doing whatever it takes in getting their child to learn to read.

The same cannot be said when it comes to numeracy.  On the topic of math, some parents will throw up their hands.  They will say they never understood math so they are not surprised their child is having difficulty.  The openness to supporting the child does not get the same emphasis as language did a few moments ago.        

I have heard of parents and grandparents going back to high school to complete their diploma’s just to be a role model to their children and end the cycle of illiteracy that kept occurring within the family.  I don’t know many stories where parents went to the same measures to show their commitment to math. 

Why is this?  Why is it that some people are not concerned with being able to make correct change but are embarrassed that they cannot read a menu or put together a proper sentence?
These questions lead me to think of our reliance on math tools and my own family background.  I come from a family of skilled trade workers.  A culture of math is ingrained in the family.  I can remember feeling a tinged of shame if my grandfather asked me an application question related to math and my estimate was no where near the mark.  As a carpenter, I never saw him pull out a calculator when he was designing or completing a project even though he put high standards on the work.  The standard he set for himself resonated throughout the family.

Our “new” methods of teaching math are the same as the methods my grandfather used in the  applications he did many years ago.  Creating an accepting culture to numeracy will lead to student success and continue to build a society that has high standards when it comes to application beyond school.

Wednesday, December 3, 2014

Why Structure is Necessary For Some

In my seemingly recent memory I was a high school student, furiously taking step-by-step notes on math rules from a chalkboard and then completing textbook work quietly in my seat (and finishing the leftovers at home). Now as a teacher of math the methods of teaching have evolved to become more student driven with an emphasis on problem solving (The Ontario
Curriculum: Mathematics Grade 9 and 10).

As I begin to plan for lessons, I’m looking for ways to make connections to my students lives in meaningful ways that will spark their curiosity about math. We take for granted that we are able to infer patterns and connections as well as be reflective, plan and self-regulate our own learning. These skills make up a person’s executive functioning skills and when a student has a learning disability that impairs their executive functioning, the modern methods of teaching Math may not be helping these students out. When executive functioning is impaired students struggle to plan, organize and self-regulate and these are arguably essential skills related to successful math completion.

How then can a teacher support students with executive functioning challenges and still create a classroom that allows for discovery? Roditi and Steinberg’s chapter “The Strategic Math Classroom” provides some sound advice. This advice is outlined below to help teachers enhance the learning for a subset of learners in four key areas:

Support Memory:
Memory strategies help enhance the automatic recall that is essential in math. Remembering the nuances and details required to solve problems can be overwhelming to young learners. Verbal strategies such as acronyms are a great way to help students remember sequences and allow students to attempt problems in a structured way. BEDMAS is one such strategy that can help with algebra. Similarly, the acronym KNOW can be used as an overall strategy (Key Words, Numbers, Operations, Work it out). Visual strategies, are another way to encode information these may include drawing cartoons or concept maps. Last but not least, hands-on strategies are a great way to consolidate information (e.g., manipulating tiles and pipe cleaners).

Organizational strategies:
Students with weak organizational strategies are easily overwhelmed and often approach problems with out a specific strategy. The students benefit from direct instructions which not only include order of operations but how to differentiate between essential and nonessential information. Having students have dedicated pages in their notebooks that outline strategy tools such as vocabulary lists, charts, tables or mnemonics are helpful and essential resources for students with executive functioning difficulty. A strategy new to me would be RAPS: reread and rephrase (promotes understanding), artistic (multiple depictions may help with problem-solving), predict (supports problem-solving) and solve (using procedural knowledge). Alternatively, students could use a Three-Column note taking technique where students use one column for each: important terms, definitions and examples. Check out page 247 in Roditi and Steinberg article for pictorial representation.

Shifting representations and formats:
As teachers we likely do this already but it’s important to note that some students need to different representations that link two concepts they already understand in order to bypass difficulties. It’s just about shifting perspectives, but not expecting students to make the leap on their own. One should be aware that subtle changes in language or format can be very challenging for students with executive function problems. This is where study guides help students practice their test taking abilities.

There are also times that students struggle to select the appropriate strategy to solve a problem. If disorganization reigns as a part of executive function problems it’s easy to see why students would have difficulty solving problems with multiple steps. Teachers can use templates with sequential steps that will help students discern between important information. An example of this is teaching linear equations. Once they’re anchored with their template students can use this to move on to more complex reasoning.

Checking strategies:
Making efficient use of time during tests and the ability to self-monitor and check work can make a huge difference in achievement. How can teachers help support those who do not have the strategies which seem innate to others? Error analysis in which the teacher works with the student to identify individualized common mistakes. They can then come up with “strategy tips” that a student can write at the beginning of the test as a means of reference for checking.
Remember that learning by solving problems without rules, routines or repetition can cause some students to feel lost in the nebulous that is learning. With just a few quick strategies implemented into teaching lessons and paradigms students with executive functioning difficulties can benefit from some of the old styles that many of us grew up with. It is not about making major changes but rather about embracing the past and mixing it with the future.
For more examples consult:

Roditi B. N. & Steinberg, J (2011). The Strategic Math Classroom. In L. Meltzer (Ed.). (2011). Executive function in education: From theory to practice (p 237-257). Guilford Press.

Tuesday, October 28, 2014

Training Kids For a World That Doesn't Exist

Our education system was built to address a world that won't (or currently doesn't) exist for our students because it is changing so rapidly. When our students graduate, they will be faced with very different demands than many of us were faced with. The education system that most current teachers grew up in is not necessarily the system that will deliver the strongest skills to our students today. We need to teach our students skills that will allow them to be successful in this ever changing world. Students will be working in careers that don't exist yet and the technology that they will likely use in their everyday life we haven't seen or heard of yet. The onus placed on the education system is huge. The expectation is to have these students ready to change and adapt as quickly as the world around them and be creative, but still have strong communication, written and oral language skills, problem solving skills and the necessary math skills among others. Here is a quick, creative youtube video that reminds us of the 21st Century Skills.

The following article discusses how some of the schools in the US are changing their way of thinking and teaching to address this ever changing concern. It speaks of the many discovery and innovation classes that are becoming more popular. They are creatively responding to discovery learning at Harvard in many disciplines such as science, engineering, and business to provide a richer learning environment that will foster deeper learning. Several other US university campuses are also using an innovative discovery approach which is now filtering into high schools, middle schools and after school programs. It suggests how powerful innovative discovery can be and how meaningful and lasting the learning is.

I am interested in hearing your thoughts about innovation, discovery, the iterative process and the role it plays in education.  What activities are you trying in your classrooms to move down this exciting road of learning?

Monday, October 27, 2014

Math is not equal to calculating

I think it’s important as Math Educators to watch the video of Mr. Conrad Wolfram who’s the Mathematician himself and the CEO and the Co-founder of the Wolfram group. Mr. Wolfram believes that the math education around the world has a problem, the Governments think their math is failing in their county, students think its difficult, teacher’s find it a hug struggle to move the mathematics of their students forward and people who want mathematics in the outside world like employers find that they don’t have enough Mathematics.

He believes we are living in a world that’s ever more quantitative and more mathematical than before however we have got falling interest in education in Math. He asks, “Why do we have this chasm between the two math” (math in education and math outside)? He believes there is one simple answer to this problem: Computers.

Wolfram points out how the math in the real world is problem solving, modelling, stimulating, thinking out what the questions are and analyzing the results but in education its doing calculating mostly by hand and if your lucky by calculator, the problems small and distance from real world. He believes that we should be trying to bring the two together to engage students.

 Mr. Wolfram explains the four steps in doing math, posing the right question about your situation, turn that from real world in to math formulation and put it in to the specific math setup to do step 3, calculating. The step 3 is taking it from your setup to answer from mathematical form. Step 4 is from that mathematical form to real world and crucially verifying it. He explains how, “Perhaps 80% of doing math education at school is step 3 by hand and largely not doing steps 1, 2 and 4.” He believes step 3 is something computers can do better than any human and as he points out we don't want our students to be third rate computers but to be first world problem solvers.

In conclusion, he recommends an open-ended use of computers and encourage those who argue that computers "dumbs math down" to look in to the real world and see how science and engineerings and other things that depends on math have got much more conceptual. Please watch this video and share your comments as I think he has some great points and suggestions!

Sunday, October 26, 2014

Why is it your job to teach your kid Math?

This article, from MacLean’s magazine, two and a half years ago, is targeting parents whose kids are not achieving in math.  That is, they don’t have the basic skills to undertake high school math and parents are now asking why?

For me as a high school math teacher, it speaks to why kids come into grade 9 math without basic computations skills.  Why are we having such low math scores?  It is not just Ontario, but across Canada.

The article explains how these gaps have developed over the last ten years or so. I particularly liked the reference to the Tower of Babel – building that tower to heaven (heaven being Math in our case) and then finding one day that no one was speaking the same language while everyone was working toward the same outcome – get these kids to Math Utopia.  The endless ways of learning numeracy skills is confusing everyone – especially K – 8 math students.  Asking them which way works best for them when they are presented with a myriad of basic computation lessons is puzzling for the developing learner.  There certainly is a case for “over kill” in the computation department.

The downside of this article is that parents can, yet again, place blame on schools.  It is best that all stakeholders work together to solve the gap conundrum in computation skills.

Saturday, October 25, 2014

Rethinking What We Teach as Math Teachers and How We Teach it

When I read Jessica Lahey's article about Steve Strogatz,Professor of Applied Mathematics at Cornell University  teaching an introductory math course for non-math majors who hate math, I am saddened.  Strogatz has his students submit a math biography outlining their math encounters throughout their life.  He shares that his Liberal Arts students have had unpleasant math experiences and they blame themselves for not understanding math, they feel they are not intelligent enough or talented enough to do math.  Strogatz (and others) teach an inquiry based math program at Cornell University to Liberal Arts students to help them see math in a different light and feel good about themselves and math. He states with the right approach he has been able to turn students views about math around.  He feels this turn around is related to how the math is delivered to these students.  The program he delivers is called Discovering the Art of Mathematics: Mathematical Inquiry in the Liberal Arts.  If you follow the previous link there are student testimonials, quotes and videos describing the experiences they have had in their Cornell math class in comparison to their prior math experiences in high school. 

I believe these messages are important messages for us as secondary teachers to hear. Sometimes I think teachers don't always stop and realized just how much they can affect their students.  Rethinking how we teach math and ensuring we do our best to reach every student using multiple means is important. Taking the extra time can put students on one future path or another, all because of the experience they have had in a classroom. Here is a video of Strogatz at Cornell with students who enjoy math and are studying Applied Math to use in pursuit of future careers.

Friday, October 24, 2014

Teaching Math through dance and movement

 In the following video, Erik Stern and Karl Schaffer decided to take their love of dance and apply it to their classrooms to teach math through dance movements. 

They discovered a few key things about how to learn, the importance of how embodying a problem is memorable, social and creative, the student’s physical energy is no longer a concern and choreography and mathematical thinking are composed of similar building blocks, remembering sequence, asking the things are bigger or smaller and check your work to see if its consistent. They also explained symmetry and mirror image with dance movements and by copying each other movement and how we tend to think more effectively with special imagery on larger scales. They want to create a mathematics classroom environment where teachers would just say, “ put the books away, and push the desks aside, lets warm up, its time to learn Mathematics”.

In a video below, students use music and sound movement to help them describe different graphs such as linear, quadratic, absolute, cubic. What I found interesting about this approach is students learning these through sounds and movement, which makes the learning more engaging and encourage learners to make their Mathematical thinking more visible. What the words mean, sound and movement help them identify different functions. In my classroom, my students use dance movements to describes different types of graphs as well as transformation concepts in grade 11 and 12. The students should create their own dance movements to describe graphs and their transformations, which is similar to the video here. 

In another video, the students learn in a reach learning environment which increase their understanding of mathematical topics such as congruence, symmetry, transformation, angles and degrees, mapping on a coordinate grid as well as deep experience with mathematical practices and problem solving. 

I do believe by the end of the lessons students have a better understanding of the concepts learned in the classroom and they learn that mathematics can be fun. I do believe bringing dance movements in to Mathematics classroom could benefit students with different learning abilities. Please share if you tried dance movement in any of your lessons.


Wednesday, October 22, 2014

Math teachers beware: New app solves equations with a camera

This morning on early morning news I saw a demonstration of this app.  I have been thinking about it all day and really only coming up with a question or two as to the implication in senior math classes – or for that matter, any math course.

This article by Michael Thomas shows that the technology is similar to taking a picture of a check to put in your bank account.  As the camera takes a picture of the question or problem, the answer is provided.

The article goes on to state obvious concerns that teachers may have to be aware about students who may use it to cheat on tests and reducing time spent thinking through questions and problems assigned for homework.

It did say that not all problems were able to be solved, but that the arsenal of problem solving capabilities will improve and expand continually.

The purpose of this new app is not trying to be sold as a way to learn – but as visionary technology.  I am not sure about the term or concept here of “visionary technology”.  It seems to be a term to fleece consumers and give math teachers more to blog about.

I do not think that math teachers have to beware about this app.  Yes, this app could be a bit more than the answers in the back of the book, but could be used the same way.  Most of our senior students would quickly learn to use the app in a helpful way.  They would know that they will not learn by just copying the answers from the app.  Senior math teachers just be aware and relax.

Multiple Ways To Multiply

As an occasional teacher, I see different students every day with very different abilities and understandings when it comes to a variety of subjects. Gaps in education, math in particular, are inevitable and as an occasional teacher who is only working with students with gaps for a short time, I have had to develop a bag of tricks to give students tips and strategies for solving problems. I find that a lot of students struggle with mental math skills and multiplication seems to stump students. When I am working with a class for only a day, it is hard for me to teach the students complex math skills; thus I try to think of strategies I can leave with them that they can take with them and apply to every area of math. So many students turn to their calculator and no longer rely on mental math skills to complete simple multiplication. When it comes to such a crucial skills, I have a few strategies I like to introduce to classes - you never know what strategy may click with a student. In my experience, visual learners really seem to enjoy the Japanese technique of multiplying with lines explained in the following video.

Gone are the days when teaching should be uniform, as education has evolved, we have come to see that each student learns in their own unique way. What better way to reach out to students and instill a love for mathematics than offering them a variety of strategies they can use to solve a problem in a way that makes sense to them. Do you have a trick that will help students with mental math? I'd love to hear from you and add it to my bag of tricks so that I can share it with my students.

Monday, October 20, 2014

The Math in Poetry

One of my favourite strategies to use when writing poems is to try to incorporate mathematics terms, formulas, and theories into the story, image, or moment I'm presenting in my poem. This comes from a deep love for mathematics and calculations, and I am not the only poet who expresses her or his love or fascination for math in this way. Take this poem "Burial" by Robert McAlmon for example where he uses mathematical principles to explore his understanding of life and death:


by Robert McAlmon1895 - 1956

Geometry is a perfect religion, 
Axiom after axiom:
One proves a way into infinity
And logic makes obeisance at command. 

Outside of the triangle, cubes, and polystructures 
There is restless pummeling, pounding and taunting. 
The end is diffused into channels
Every step into eternity—and steps are endless.

There's even an entire movement of poetry devoted to applying mathematical principles to poetry in order to explore the potential of poetry within specific restraints and formulas: OULIPO. For example, you might create a new poem by applying n+7 (i.e. replacing every noun in the poem with the noun that comes seven entries after it in the dictionary). 
 In my own attempts to connect math and poetry, I incorporated poetry into my grade 7 teaching practicum by asking students to create concrete poetry (where the image/placement of the words is as essential to the meaning of the poem as the words themselves) that demonstrates knowledge of translations, reflections, and rotations. Here's an example of one student's work (that I thought showed nice understanding of the concept):
 Lastly, I think Slate has a very good idea right here. I am completely embracing this next April.

Wonder Shelf In Math


I am so inspired when I read Rafranz Davis' article about providing her math students with a shelf of manipulatives and tools that afford them the opportunity to express creative freedom and ask wonder questions.  She calls this shelf a Wonder Shelf and it has been many years in the making. She includes simple household items from the kitchen, toys, blocks and lego pieces, arts and craft items as well as a variety of forms of technology. Read about at,

Her students are able to access this shelf during class and before/after school not only to extend their learning in areas they are studying, but also to reach out and explore other topics. I can only imagine her surprise when she turned on an ipad to find stop motion clay figures demonstrating changes in volume of a cylinder, when this was not even assigned.  A student took it upon themself to create the clay figures and produce the stop motion clip possibly as a way to help them understand the concept being taught in class.  How exciting for a teacher to make this discovery!  Students can surprise us in so many ways when they are given the freedom and space to be creative.  I'm certain this particular student will remember changes in volume of a cylinder for many years to come because the memory of creating the stop motion clip in Ms Davis' class will stick with them forever!

Math Brain vs English Brain

by Julie Homenuik

As an English major and teacher, I hear a lot of negativity about mathematics: “We’re English teachers—we can’t/don’t do math.” “Math can stay over there.” “Ugh. Math.” “You like math?” “I’m good at English, but not math.” To which I say, “Really?” I don’t believe that math and English are as disparate as some people make them out to be. So, with only a slight confirmation bias, I went to the internet to prove my claim that math and English are not opposing ends of a subject-spectrum.

One observation I have made in my teaching career is that students that do very well in English usually also do very well in math. Actually, the grades that students get in English are usually very similar to their math grades, even when they tell me that they are very good in English, but not very good at math (which in itself is an interesting perception). For this blog post, I wanted some objective evidence to validate my observations, and I found one interesting infographic based on SAT statistics that suggests that math and verbal scores on the SAT are comparable. According to the stats presented here, very few students had completely divergent scores in math and language.

The idea that math skills and English skills are divergent seems to relate to the Left Brain/Right Brain myth. People fall back on the right brain/left brain theory as if we simply work with one side of our brain when tackling certain subject areas. And then, to add insult to injury, some decide that math is associated with left brain thinking and language arts with right brain thinking. It’s absurd to think that math does not incorporate creativity or English class does not endorse logical, analytical thinking.  Luckily, the right brain/left brain theory is simply a myth that we simply like to believe because it lets us easily categorize our skills (and sometimes, I think, excuse our perceived weaknesses).

I actually see math and English as having a fair bit in common. Math is a language after all. In both math and writing, we abide by rules, order, structures, and symbols. We must interpret and analyse the information in front of us. We use numbers, rhythms, and patterns in writing just as we look for in mathematics. The skills we use in both subject areas are not divergent, but rather interconnected. For another perspective on this, I direct you to an article by Melanie Carbine, “Math and English: More in Common than Different”. She too notes the connection between math and English language.

Probably the best piece of information I came across is that having a strong English teacher actually makes students stronger in mathematics. Stanford researchers determined that “students of good language arts teachers had higher than expected math scores in subsequent years” (“Stanford research shows long-run benefit of English instruction”). I would think, then, that being strong in English also leads to stronger skills in mathematics—which can also be seen in the SAT stats as well since so few students who scored well in verbal did poorly in math. Therefore, with regards to all my English major math-naysayers who claim they are math-impaired, I say they are actually well-equipped to tackle mathematics (so stop dissing it!)